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Thursday, 2 August 2018

Quantification ∀

Universal Quantification ∀

∀ means "for all"

Allows us to make statements about all Objects that have certain properties

Can now state general rules:
 - ∀ x King(x) ⇒ Persons(x)
 - ∀ x Person(x) ⇒ HasHead(x)
 - ∀ i integer(i) ⇒ Integer(plus(i,1))

Note that
∀ x king(x) ∧Person(x) is not correct!
This would imply that all objects x are kings and are People
∀ x kings(x) ⇒ Person(x) is the correct way to say this

Existential Quantification ∃

∃ x means "there exists an x such that...." (at least one object x)

Allows us to make statements about some object without naming it

Examples:

∃x king(x)
∃x Lives_in(John, Castle(x))
∃i integer(i) ∧ GreaterThan(i,0)

Note that ∧ is the natural connective to use with  ∃
(And ⇒ is the natural connective to use with ∀)

Combining Quantifiers

∀ x ∃ y Loves (x,y)
 - For everyone ("all x") there is someone ("y") who loves them

∃ y ∀ x Loves(x,y)
    - there is someone ("y") who loves everyone
Clearer with parentheses: ∃ y (∀ x Loves (x,y))

Connections between Quantifiers

Asserting that all x have property P is the same as asserting that does not exist any x that don't have the property P

∀ x Likes(x, 271 class) ⇔ ᆨ∃xᆨ Likes (x,271 class)

In effect:
 - ∀ is a conjunction over the universe of objects
 - ∃ is a disjunction over the universe of objects Thus, DeMorgan's rules can be applied

De Morgan's Law for Quantifiers


Rules is simple: If you bring a negation inside a disjunction or a conjunction, always switch between them (or⇾and, and⇾or)

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